The last coin has a more exacting duty than the first. The first may answer a need so urgent that nothing else comes near it. The last must justify its place against every other use still open to the purse. If a little less wine would buy bread that matters more than the wine surrendered, the arrangement can be improved without another penny of income. Nothing in the world need become more plentiful; the purchaser need only move what he already commands. The equimarginal rule is the condition under which that movement has finished its work. Each small addition to an active use of expenditure must yield as much as the addition forgone elsewhere, or the purse has not yet been made to do all that it can.
The earlier Equimarginal presented this as the master rule of allocation. Its starting point is almost embarrassingly modest: do not leave an available improvement unused. Jevons considered a fixed stock of a commodity capable of serving different purposes. If a little barley contributes more as food than in another use, moving it toward food improves the distribution; where both uses remain worthwhile, the transfer stops when the final contributions agree. Money extends the comparison by buying different quantities of different goods. A coin spent on bread and a coin spent on wine are equal claims upon the budget. A loaf and a bottle need not be. It is the return to the common resource surrendered that has to be compared.
Suppose, then, that bread costs twice as much per unit as wine. An additional loaf must contribute twice the marginal utility of an additional unit of wine for the last pound spent on each to have the same return. To demand equal utility from the two physical increments would be to overlook their different prices. In the familiar notation, the interior condition is Ub/pb = Uw/pw. It compares the marginal gains available at the same chosen bundle. Move to a different bundle and the comparison may change. Alter income and the whole arrangement may change. The equality does not confer a permanent equivalence upon bread and wine; it describes a particular balance reached under particular terms.
A small invented purse will make the distinction visible. Give the purchaser twelve pounds, price bread at two pounds and wine at one, and represent his ordering of positive bundles by U = log b + log w. Four units of each exhaust the purse. But three units of bread and six of wine also cost twelve, and the second bundle is preferred: its index is log 18 rather than log 16. The improvement does not arise because the quantity of everything has risen. Bread has fallen and wine risen. At the first bundle the extra pound directed to bread has a marginal return of one-eighth, while the pound directed to wine has a return of one-quarter. At the second, both returns are one-sixth. The shift has brought the rival uses into balance.
Notice how little the result says about the purchaser’s inner arithmetic. He need not recite a logarithm before breakfast. A formal representation of his choices is not a report of the procedure by which he arrives at them. He might reach the bundle through habit revised after disappointment, repeated comparison, advice, or an explicit calculation. The model, as stated, does not choose among these explanations. It says that the final arrangement admits no better affordable alternative according to the ordering it assigns him. To infer from that representation that he carries out the economist’s calculation would be to mistake a description of the destination for an account of the journey.
At an optimum, no further transfer improves the choice. The equation need not be a calculation inside the chooser’s head.
Even the apparent weighing of sensations is less demanding than it first appears. In the example, replace the logarithmic index by V = bw. Every pair of positive bundles retains its old order, because exponentiating the first index gives the second. At the best bundle the bread and wine margins now read six and three; divided by their prices, both are three. The former common value of one-sixth has disappeared, but the equality remains and the shopper takes precisely the same basket. More generally, a differentiable transformation with positive derivative multiplies all the marginal utilities at a given bundle by the same positive factor. Their ratios survive. The balance needs the rate at which the purchaser is prepared to substitute one good for another; it need not read the amount of pleasure on an absolute scale.
But an equality can be a remarkably misleading badge of excellence if the conditions that earn it are forgotten. Where the goods are divisible, the ranking smooth and the chosen amounts positive, a small transfer can be made in either direction, and the familiar first-order condition is necessary. It becomes sufficient under suitable shape conditions, such as a concave utility representation over a convex feasible set. Without them, a stationary point can be the worst affordable basket. Change our invented purchaser’s index to b² + w² while keeping prices and budget unchanged. At b = 4.8 and w = 2.4, both marginal returns per pound are 4.8. Yet that bundle gives 28.8, while spending everything on wine gives 144. The equality holds at a minimum along the budget. The last coin is balanced because either direction would improve its deployment.
The other common exception is less dramatic and more ordinary in form: the best allocation may give nothing to one use. Imagine a further invented ordering, U = 2b + w, with bread priced at one pound, wine at two, and six pounds to spend. Every pound buys two index units through bread and only half a unit through wine. All six pounds go to bread. The margins never become equal, and no failure has occurred. At a boundary the purchaser can transfer expenditure into the unused good but cannot remove expenditure that is not there. The appropriate test asks whether that available move would help; it does not require equality with a move the boundary forbids. Jevons allowed precisely the possibility that one use continued to outrank all the others even when nearly the entire stock had been assigned to it.
There is a further literal difficulty with the last coin: some things cannot be bought a coin’s worth at a time. A ticket, a machine, a journey undertaken at all may come in lumps. Then the choice is between feasible packages, and their differences must be compared as packages. An infinitesimal saving that buys nothing cannot be treated as if it already purchases a small portion of the alternative. Nor can a household release the whole of a supposedly discretionary pound if doing so breaches a commitment that formed part of the choice. The general principle—seek the best available allocation—survives these cases. The neat marginal equality is one expression of it, available only where the feasible alternatives admit the movements the equation imagines.
What, then, must a person be for the last coin on bread to weigh exactly the last on wine? He must, within the model’s domain, have an ordering that supports the relevant trade-offs, face the stated prices and constraint, choose positive quantities, and arrive at an allocation satisfying the appropriate optimality conditions. He need not possess a private instrument for reading units of satisfaction. Whether actual choices have the consistency and stability required by such a representation is a further question, answerable by examining choices and constraints rather than by admiring the equation. The power of the rule lies in exposing an available improvement. Its overreach begins when the absence of one in a model is taken as proof that a flesh-and-blood purchaser has searched every possibility.
A fixed purse thus acquires a theory of its own. The preceding plate distinguished the useful whole from the increment; this one has made the increment answer to the opportunities it displaces. Neither rule can determine the terms offered by the world outside the household. Bread and wine arrive with prices attached, and the buyer’s balancing acts take those prices as given. To explain the price itself requires following the other side of the trade, where production and supply alter what can be offered. The next plate brings the two sides together. The purchaser has found his balance; the market has yet to find its own.